The relationship between confidence intervals and hypothesis testing In an effor

Question

The relationship between confidence intervals and hypothesis testing In an effort to better manage his inventory levels, the owner of two steak and seafood restaurants, both located in the same city, hires a statistician to conduct a statistical study. The owner is interested in whether the restaurant located on the south side sells more filet mignons per night than the restaurant located on the north side of the city. The statistician selects a random sample of size n_1 = 35 nights that the southside restaurant is open. For each night in the sample, she collects data on the number of filet mignons sold at the southside location and computes the sample mean M_1 = 9.09 and the sample variance s_1^2 = 28. Likewise, she selects a random sample of size n_2 = 32 nights that the northside restaurant is open. For each night in the sample, she collects data on the number of filet mignons sold at the northside location and computes the sample mean M_2 = 7.49 and the sample variance s_2^2 = 27. The statistician checks and concludes that the data satisfy the required assumptions for the independent-measures t test. Then she computes the 95% confidence interval for estimating the difference between the mean number of filet mignons sold per night at the southside restaurant and the mean number of filet mignons sold per night at the northside restaurant. This 95% confidence interval is 1.60 plusminus 2.5624 filet mignons. If she were to formulate null and alternative hypotheses as H_0: mu_1 - mu_2 = 0, H_1 : mu_1 - mu_2 notequalto 0 and conduct a hypothesis test with alpha = .05, the null hypothesis ______ rejected based on the result that a difference of zero ______ in the computed interval. Hence, she would conclude that there ______ a significant difference between the mean nightly sales of filet mignons between the two restaurants.

Explanation / Answer

Solution:

For the given confidence interval we are given

Confidence interval = 1.60 -/+ 2.5624

Lower limit = 1.60 – 2.5624 = -0.9624

Upper limit = 1.60 + 2.5624 = 4.1624

Confidence interval = (-0.9624, 4.1624)

For checking the null hypothesis, the difference of zero is not included in the above confidence interval, so we cannot reject the null hypothesis.

Answer:

If she were to formulate null and alternative hypotheses as H0: µ1 - µ2 = 0, H1: µ1 - µ2 0 and conduct a hypothesis test with = .05, the null hypothesis is not rejected based on the result that a difference of zero is included in the computed interval. Hence, she would conclude that there is not a significant difference between the mean nightly sales of filet mignons between the two restaurants.

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